Measurements of surface elastic torques in liquid crystals : a method to measure elastic constants and anchoring energies

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Measurements of surface elastic torques in liquid crystals : a method to measure elastic constants and

anchoring energies

S. Faetti, M. Gatti, V. Palleschi

To cite this version:

S. Faetti, M. Gatti, V. Palleschi. Measurements of surface elastic torques in liquid crystals : a method

to measure elastic constants and anchoring energies. Revue de Physique Appliquée, Société française

de physique / EDP, 1986, 21 (7), pp.451-461. �10.1051/rphysap:01986002107045100�. �jpa-00245463�

Measurements of surface elastic torques ⁱⁿ liquid crystals :

a method to measure elastic constants and anchoring energies (*)

S. Faetti (+°), ^{M. Gatti} (+) and V. Palleschi (+)

(+) Dipartimento di Fisica dell’ Universita’ di Pisa, ⁵⁶¹⁰⁰ Pisa, Italy

(°) Gruppo Nazionale di Struttura della Materia del CNR, ^Piazza Torricelli 2,56100 Pisa, Italy (Reçu ^{le 23} septembre ^1985, révisé les 6 décembre 1985, ^et 7 avril 1986, accepté le 7 avril 1986)

Résumé. 2014 Le couple exercé par

cristal liquide nématique

les surfaces est mesuré

utilisant

pendule ^de

torsion. Ce couple ^est engendré par

champ magnétique qui provoque

distorsion du directeur. Trois géo-

métries différentes sont étudiées. Les constantes élastiques ^et l’energie d’ancrage du cristal liquide nématique peuvent ^{être obtenues} par la

du couple. L’avantage ^{de cette méthode est} que la

des constantes élas-

tiques ^est peu sensible

petites desorientations du directeur près ^{des surfaces et}

^{valeur finie de} l’énergie d’ancrage. ^Nous

^{utilisé cette} technique pour la

des constantes élastiques K11 and K33 ^{du cristal} liquide 4-pentyl-4’-cyanobiphenyl (5CB). Les résultats de l’expérience ^sont comparés

^des

précé-

dentes.

Abstract

Surface torques ^exerted by

^nematic liquid crystal

^solid plate

investigated. Three different

geometries

considered. The torques

generated by applying

^uniform magnetic ^{field to} the nematic sample

and

measured by

^torsion pendulum. ^{The elastic constants of the nematic} Liquid Crystal ^{and the} anchoring

energy ^at the interface

be obtained by this kind of measurements. As

main advantage ^{of this} technique, ^the

measurements of the elastic constants

poorly ^{sensitive to small} misalignments of the director and

not

affected from

finite value of the anchoring energy. Experimental results for the splay ^and bend elastic constants of the nematic LC 4-pentyl-4’-cyanobiphenyl (5CB)

given ^and compared ^with previous experimental ^results.

Classification

Physics ^Abstracts

61.30

62.20D - 68.1OC

1. Introduction

Elastic constants of nematic liquid crystals (LC) play

an

important ^{rôle on the} macroscopic properties ^of

these materials. As

consequence of this,

^{lot of} experiments ^{have been} performed ^{to measure elastic}

constants of nematic LC [1-13]. ^{Most of the} experi-

ments concern the investigation ^{of director deforma-}

tions induced by magnetic ^or electric fields in thin

layers of nematic LC [1-8]. ^Other experiments ^concern

the investigation ^{of the} light scattering ^{from an uni-} formly aligned layer of nematic LC [9-11 ]. These latter

experiments

^affected by larger uncertainties. Other methods have been proposed ^but they ^{do not seem to}

fumish

sufficient accuracy [12, 13]. ^{Therefore so far}

the most reliable values of the elastic constants have been obtained from measurements of the Freedericksz transition in thin nematic layers. ^{In most of these}

(*) ^Research supported ⁱⁿ part by ^{Ministero della Pub-}

blica Istruzione and in part by Consiglio ^{Nazionale delle} Ricerche, Italy.

experiments ^the magnetic (or electric) ^{field is} applied perpendicular ^to the director and

director distortion

occurs when the field exceeds a threshold value Hc.

By assuming strong anchoring ^{of the director at the}

interfaces of the nematic layer ^{one obtains} [14]

where d is the thickness of the layer, xa is the aniso- tropy ^{of the} diamagnetic susceptibility ^and Kii ^{is an}

elastic constant (i

¹ splay, i

² twist, i

³ bend).

The three values of the index ⁱ correspond ^to three diffe- rent experimental geometries [14]. Rapini ^{and Papou-}

lar [15] investigated ^the sensitivity of the threshold field to small misalignments of the director and to a

finite value of the anchoring energy. They ^{found both}

these factors reduce the threshold field, ⁱⁿ particular ^a 2°-misalignment of the director at the surfaces gives ^a

reduction of 10 % in the threshold magnetic ^{held This} high sensitivity ^{to the} experimental conditions ^can

explain ^some large discrepancies ^{between different}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01986002107045100

measurements of elastic constants. We note, however, that in recent papers [6] Oldano et al. proposed

«

modified Freedericksz technique » ^{which avoids the} previous error sources and fumishes accurate values of the elastic constants. In the following ^we will denote the conventional Freedericksz method as

standard » Freedericksz technique ^{and the} method of reference [6]

as «

modified » Freedericksz technique.

In a recent paper Grupp [16] proposed

^new expe- rimental method to measure the twist elastic constant of nematic LC. The method consists in measuring ^the

elastic torque ^exerted by

twisted nematic layer ^on

the surfaces. The two parallel plane glass plates ^which

bound the nematic layer ^are treated in such a way ^{as to} induce a planar orientation of the director. The twist of the director-field is generated by rotating ^one

of the two plates. ^The main drawback of this method is related to the long measuring ^time ( N ^a day) ^due

to viscosity ^{effects. In}

^recent paper [17] ^{we showed}

that the measuring ^{time can be} largely ^reduced (to

few minutes) using thick nematic samples ^and gene-

rating the twist of the director field by ^{means of}

magnetic ^held.

In this _paper we show that the experimental ^method

of reference [17] ^{can be} suitably ^{extended to measure}

the bend and splay ^{elastic constants} of nematic LC.

Three different geometries ^are investigated ^which

allow us to measure the three elastic constants K11, K22 ^and K33. ^{The main} advantages and drawbacks of this new experimental technique ^are discussed The bend and splay ^{elastic constants} of the nematic LC

4-pentyl-4’-cyanobiphenyl (5CB) ^are measured and the experimental ^{results are} compared ^with previous

results obtained by other authors.

2. Principle ^{of the} measurement and experimental apparatus.

Figure ^{1 shows} schematically ^the experimental appa- ratus used to measure surface elastic torques. ^The nematic LC sample (NLC ⁱⁿ Fig. 1) ^{lies on the bottom}

of a cylindrical glass cell which is thermostated by ^a

water thermal bath (T) ^with

temperature stability

of 10 mK. The temperature is measured by ^{means of a}

linear thermoresistor (TR ⁱⁿ Fig.1). ^A glass plate (P) ^is dipped ^{in the nematic} sample ^{and is sealed to a vertical} glass tube (Q). ^{Three different} configurations ^{are inves-} tigated (i

^{1, 2, 3 in} Fig. 2). ^The glass plate ^{is sus-} pended by

^thin quartz ^wire (W) through ^the glass

tube Q. The torsion pendulum ^can be rotated around the vertical axis by ^means of a rotation stage (R) ^which

lies on the top ^{of the} cylindrical glass cell. An uniform

magnetic ^{field H} ranging ^{from 50 to 8 800 G can be} applied along the horizontal x-axis. When the _easy axis of the director

the glass plate ^{is not} parallel

to the magnetic field, ^a distortion of the director-field

occurs within a thin layer ^{close to} the surface (see Fig. 3) (the thickness of this layer is of the order of the

Fig. ^1.

^{Schematic vertical}

section of the experi-

mental apparatus. ^NIC

^nematic sample, ^P

glass plate (the geometry ⁱ

^{2 of} Fig. ^{2 is} shown), Q

vertical thin

glass tube, ^M

glass plate (used

^{mirror to detect the}

rotation of the torsion pendulum), ^W

^thin quartz ^wire (30 gm), ^R

^rotation stage, ^T

thermostatic bath of

circulating ^{water, TR}

^linear thermoresistor, ^H

magne- tic field

Fig. ^2.

^{The three} geometries ^{of the} glass plate ^P of figure ¹

and the corresponding director orientation at the surface

are

shown : i

1, the easy axis of the director

is parallel

to the surface of the rectangular glass plate (homogeneous orientation); ⁱ

2, the easy axis of the director is parallel ^to

the surface of the circular glass plate; ⁱ

3, The easy axis of the director is orthogonal ^{to the} rectangular glass plate. f3 ^is

the angle between the magnetic ^field and the director at the surface of the glass plate.

magnetic ^cohérence length 03BE

Kii ~03B1 1 H ^{and ranges}

from _{2 lim} to 20 _lim in our experiment). Under these conditions the nematic sample ^exerts

torque ^{on the}

glass plate ^{P and} thus, the torsion pendulum ^rotates.

Fig. ^3.

Schematic view of the director distortion

the

glass plate. a) ^{Cases i = 1} (~

90°) ^{and i}

³ (ç

00) :

horizontal

section of the glass plate ^P.

^{is the} angle

between the axis y’ orthogonal ^{to the} plate ^{and the} y-axis orthogonal ^{to the} magnetic field, ~ is the angle ^{between the}

director at the surface and the y’-axis. ^The director-field lines in the presence of the magnetic ^field

shown schemati-

cally. ^For convenience the director-field has been drawn only

in the lower region ^{of the} figure. ^A symmetric distortion of the director

in the upper region ^{above the} glass plate.

b) ^{Case i}

² (twist distortion) : ^vertical

section of the circular glass plate ^P. The easy axis of the director lies

the surface of the glass plate and makes the angle ao with the

y-axis.

This rotation is detected from the deflection of a laser beam which is reflected by ^{a small} glass plate ^{M sealed}

to the glass ^tube Q. The accuracy of the measurement of the rotation angle Da of the torsion pendulum ^is 10-4 rad The restoring torque of the torsion wire is :

where k is the torsion coefficient of the quartz ^wire which is measured with ^an _accuracy better than 0.5 %.

The elastic torque ^exerted by the nematic LC on the

glass plate ^{P can} be calculated by using the elastic

theory ^{of LC} [14]. ^{Let S’ be} the surface of the glass plate and let S" be

ideal closed surface which bounds S’and lies far _away from this surface where the director is uniformly aligned along ^the magnetic

field. The elastic surface torque ^{on the} glass plate ^is

balanced by ^the magnetic torque acting ^{on the volume}

bounded by ^{the surfaces S’and S ", tue.}

where

is the director. The dependence ^{of n on} spatial

coordinates is obtained by minimizing the total free

energy 3 ^{of the} system [14] :

where

is the director. W(~0) ^{is the} anchoring

energy ^on the glass plate [14] ^and (po is the angle

between the director and the _easy axis at this inter- face. As

first approximation ^the anchoring energy is usually ^{assumed to have the} simple ^form [ 18] :

where Wo ^{is the} anchoring energy coefficient. The Frank-Ericksen elastic constant K24 [19] ^{is not}

included in equation (4) ^{since it does not contribute}

to the free _energy in the ^case of the two-dimensional director distortions shown in figure ^3. Nehring ^and Saupe [20] extended the Frank-Ericksen theory by introducing

^{new elastic term} proportional ^{to the}

elastic constant K13. ^{This term does not influence}

the bulk free energy but it modifies the surface free energy. In ^{a recent} paper Barbero and Oldanc show- ed [21 that ^the presence of this elastic surface terms induces some difficulties in obtaining ^{the correct} boundary conditions for the director-field. In particu-

lar all previously proposed boundary conditions are

incorrect (see references given in reference [21]). ^Since

the mathematical and physical problem of accounting

for the effect of the surface-like elastic constant K13

has not yet ^been solved, ^{we here} neglect the effect of this latter constant (K13

0).

In our experiment ^the thickness 03BE of the distorted

layer ^{close to} the surfaces is much smaller than the other characteristic lengths ^{of the} system (thickness

of the glass plate (~ ^10-2 cm) ^and average thickness of the nematic sample (~ ^0.3 cm)). Furthermore the thickness of the glass plate ^{is so} small that boundary

effects can be neglected Therefore the angle ^{0 which the}

director makes with the magnetic ^{field can be assumed}

to depend only ^on the distance from the glass plate (this distance is denoted by

in the twist geometry (i

2) ^and by y’ in the other cases). ^The angles ^03B8

and _~0

obtained by minimizing ^{the free energy} (Eq. (4)) ^with respect ^to variations of 0 and _~0. Once 0 is found it can be substituted in equation (3) ^to

obtain the surface elastic torque

^{In the case i}

²

(twisted geometry ^of Fig. 3b) ^{one obtains :}

where ao is the angle between the easy axis of the direc- tor and the y-axis orthogonal ^{to the} magnetic ^field

and S is the total area of the two plane surfaces of the

glass plate. lpo is the angle of the director at the surface with respect ^{to the} easy axis. Equation (7) represents the boundary conditions which must be satisfied by

the surface director angle (po. According ^{to this} equa- tion the elastic torque acting

the director at the surface is balanced by the surface anchoring torque

- ~W ~~0). ^{Note that, for a given} value of the torque i,

the surface angle ~0 is almost zero if Wo » K22 xa

H cos (a). Therefore for

low enough magnetic field,

one can assume ~0

0 in equation (6) (strong anchoring ^{at the} interface). ^The signs + stand for the two cases 0 _ao 03C0 and - n _ao 0, respecti- vely. Two different director distortions having oppo- site twists correspond ^{to the two} signs ⁱⁿ equation (6).

If ao

0 both these distortions have the same free energy and thus, ^one expects that altemate domains

separated by ^walls [22] ^occur giving

almost vanish-

ing torque ^{on the} glass plate. ^For ao :0 0 only ^one

of these two distortions is stable and the equilibrium

surface torque ^has

^{well defined} sign.

In the case of the geometry ⁱ

¹ (or ⁱ

3) (Fig. 3a)

we

must consider that the orientation of the director

depends, by ^the symmetry ^{of this} system, only ^{on the}

distance y’ ^{from the} glass plate (n

n(y’)). ^Therefore

we

can assume the director

lies in the x’, y’ plane ^i.e.

n ⁼

(nx, n’y, ^{0) = (sin b, cos 03B4, 0),} where ô is the angle

between the director and the y’-axis. The total free energy per unit surface is

where il = K11 - K33 K33 ^{is the} anisotropy of the elastic constants, X. ^{is the} diamagnetic anisotropy, K33 ^{is the}

bend elastic constant, W(~0)

Wo ^sin2 ~0 is the

anchoring energy function, (po is the angle ^between

the director at the surface and the easy axis and

is the

angle between the unitary ^{vector k} orthogonal ^{to the} glass plate ^{and the} y-axis orthogonal ^{to the} magnetic

field

From equation (8) ^one obtains the Euler-Lagrange equation (9) ^for (fJ(Y’) ^{with the} boudary ^condition (10)

From equation (9)

^{can deduce} ~03B4/~y’ :

which can be substituted into equation (10) ^to give ^the boundary ^condition

where

have defined the surface director angle

(p

03B4(0) (see Fig. 3a). By substituting equation (11)

into equation (3) ^{one obtains} (for fi 0) :

where S is the total area of the two planes surfaces of the glass plate ^{P of} figure ^{1. The} signs ^{+ and} ± ⁱⁿ

equations (12) ^and (13) stand for the two cases 0

a

+ ~ 1800 and - 1800

+ ~ 0, respecti- vely. ç

900 in equations (12) ^and (13) corresponds

to the geometry i

¹ (homogeneous alignment), whilst qJ

⁰⁰ corresponds ^{to the} geometry ⁱ

³ (homeotropic alignment).

In the case of low magnetic fields (Kii X03B1

H cos (a) ^« Wo), ^the angle of the director at the inter- face is not modified appreciably by ^the magnetic ^field

and thus, the surface torque ^is

linear function of the

magnetic ^field (see Eq. (6)) ^{to Eq.} (13)). ^{The measure-}

ment of

for the three geometries ^of figure ^{2 allows}

to obtain the elastic constants K11, K22 ^and K33.

In particular K22 ^{is obtained} by measuring the surface torque which corresponds ^{to the} geometry ⁱ

²

(Eq. (6)). ^The procedure ^{to obtain} K11 ^and K33 ^is

more complex since, ^{in this case} (Eq. (13)), the surface torque ^is

function of both K33 ^and q. il ^can be obtained by measuring ^the ratio of the torque i (i

1)

to the torque

(i

3). ^In fact this ratio is a function

of il only. Once q ^is measured, K33 ^{can be obtained} by substituting ^the il-value ⁱⁿ the theoretical expression ^of

i

(i

1) ^{or of}

(i

3). ^This procedure gives large

errors on il. In order to clarify ^this point

^{make a}

power expansion of the surface torque ^of equation (13)

in terms of the ~ parameter ^{for the two cases i}

¹ (cp

^900,

^{90°) and i}

³ (cp

00,

0°).

After

straightforward calculation

obtain

and

Therefore the relative difference of these surface torques ^is

which is

small value since ’1 ^is usually ^{small. For} example, ^{in the case} of the nematic LC 5CB the _expe- rimental value of q ^is

0.24(1) ^and thus, 0394 ~ - 4 %.

As a consequence of this a 5 % uncertainty ^{on the}

measurement of the surface torques

(i

1) ^and

i

(i

3) ^can give ^errors greater ^{than 200} % ^on ’1.

Therefore the torque measurements do not give

accurate values of the splay ^{elastic constant} K11.

The values of K33 and il could be also obtained from the dependence ^{of the} torque ^{on the} angle

^for

a

given experimental arrangement (i

^{1 or i}

3).

This latter procedure gives, ^{in way of} principle, ^more

accurate results but it has not been used in the present experiment because of the _presence of some spurious

effects which will be discussed in section 3. 1.

So far we have assumed that the magnetic ^{field is}

small and it does not modify appreciably the surface director angle «po - 0). ^{In this case} the elastic torque is

linear function of the magnetic ^field. However, ^if the magnetic field is increased enough, ^{it can} modify

the surface director angle. Therefore the elastic torque is no yet ^a linear function of H (see Eq. (6) ^to Eq. (13)).

By looking ^at equation (6) ^to equation (13) ^{we find}

that this occurs when H z

2 W ° . This latter condition is accomplished ^{when the} magnetic ^cohe-

rence

length 03BE = Kii ~03B1 ^{H becomes} comparable ^{to the} extrapolation length ^{03B4 ~} Wu. ^{If the non linear}

regime is reached both Kii and W0 ^{can be} obtained in a

unambiguous way by the best fit of the experimental dependence of the surface torque ^{on the} magnetic

field (see Eqs. (6) ^to (13)). ^Therefore the surface torque

measurements allow to obtain, ⁱⁿ principle, ^{both the}

elastic constants and the anchoring energy coefficient.

Evidence for large deviations of i/S ^{from a linear}

behaviour has been reported by ^{us in a recent} paper(23)

where the azimuthal anchoring ^energy coefficient of the SiO-nematic interface was measured The ^accu- racy of the anchoring energy measurements depends greatly ^on the values of the angles ao and

Consider,

for instance, the twist case (Eqs. (6) ^and (7)). ^The

surface torque depends ^on the director angle ^{at the}

surface through ^a cosine function which is much ^more sensitive to variations of the surface director angle ~ when

900. Therefore the measurements of anchor-

ing energy ^{are more accurate} if ao is close ^to 90°,

whilst the measurements of elastic constants

more accurate if ao is small enough.

The main advantages ^{and drawbacks of the} torque

measurements with respect ^{to the}

standard » Free- dericksz transition measurements are :

Advantages : 1) torque measurements are poorly

sensitive to small misalignments of the director at the surface if the angle between the magnetic field and the director is close to 90°. Under these conditions, ^for example, ^{a 2°} uncertainty ^on the director orientation

gives

^{relative error on the} torque lower than 0.1 %.

This conclusion holds also if the director misalign-

ment is not the same all over at the interface. This

can be easily understood if one looks at equation (3)

which gives the surface torque

^This equation ^shows

that the total surface torque ^comes from the superpo- sition of the contributions of the magnetic torques

acting ^on the different regions of the nematic sample.

In particular ^the largest contribution to the magnetic

torque ^comes from these regions ^{of the} sample ^where

the angle 0 between the director and the magnetic

field is

450, whilst the minor contribution comes

from the regions where 0 - 90° or e

0°. Therefore, if the surface angle ^{is close to 90°, the total} torque ^is poorly ^{sensitive to} details of the surface arrangement.

2) ^{A weak} anchoring energy ^can be evidenced, ^{in a} unambiguous way, by looking ^{at the} linearity ^{of the} magnetic held-dependence of the surface torque. ^The best fit of the experimental results allows to obtain both the elastic constants and the anchoring energy coefficient. Therefore

weak anchoring ^{of the director}

at the interfaces does not affect the _accuracy of mea- surements.

Drawbacks : 1) ^A large ^amount of nematic sample

need to perform this kind of measurements (a ^few cm3), 2) ^An uniform orientation of the director is required

on a large ^surface region (~ 3 cm’). 3) ^{The measu-}

rement of the K11 ^{elastic constant} by ^{means of the}

torque technique ^{is affected} by ^a large uncertainty.

This latter is just ^{the most} important drawback of the present technique.

The main error sources of the torque technique ^{are :} a) ^the inaccuracy ^{of the area S of the} glass plate (~ ¹ %). b) ^The inaccuracy of the torsion coefficient of the quartz ^{wire W} (~ ^0.5 %). c) ^The inaccuracy

of the rotation 039403B1 of the torsion wire (~ ^{10- 4} rad).

This gives ^{a relative error on the} torque

of about 1 %

in the present experiment. d) ^The inaccuracy ^{due to}

the presence of ^a spurious background (~ ^1-2 %).

This contribution will be discussed in section 3.1.

e) ^The inaccuracy ^{of the} magnetic ^field (~ ^0.5 %).

f) ^The inaccuracy ^{due to the} boundary effects which have been neglected ^{in order to obtain} equations (6)

to (13). The maximum contribution due to this source

of error is expected when the director on the small lateral surface of the glass plate is oriented along ^the

same easy axis of the ^two plane surfaces of the glass

plate. ^{In this case a} relative contribution to the torque

of the order of 0394S/S ^is expected, ^{where AS} represents

the total area of the lateral surface. In the present experiment Ai ^1.5 %. By considering ^{all these}

error sources

estimate

maximum relative error

the parameter ^A

03C4 S.H ^{of about 5 %. The} correspond- ing uncertainty ^of K33 ^and K11 ^{are estimated to be}

20 % ^{and 50} %, respectively. ^These large errors come

from the large uncertainty ^{of il.} The accuracy of K33

can be increased to 10 % ^{if the} il-parameter ^{is known} from independent measurements. In this case K33

can be obtained from measurements performed ^either

in the geometry ⁱ

¹

^{in the} geometry ⁱ

^{3. Notice} that the uncertainty of n poorly affects the _accuracy of K33 particularly ^{in the case of the} geometry ⁱ

¹

(see Eq. (14)).

3. Expérimental ^results.

3.1 EXPERIMENTAL PROCEDURES. - In this section

we report ^some experimental ^results conceming ^the geometries ⁱ

^{1 and i}

^{3 of} figure ^2. Experimental

results for the twisted geometry (i

2) ^have already

been reported ⁱⁿ

previous paper [17]. The nematic

sample ^is 4-pentyl-4’-cyanobiphenyl (5CB) produced by BDH which has the clearing temperature Tc =

35.3°C. The homeotropic alignment ^{on the} glass plate (i

3) is obtained by dipping ^the glass plate ⁱⁿ

a

10 % water-solution of alkyl ^{benzene sulfonate} (RBS)

at the temperature ^T

^{70 OC in a} ultrasonic bath.

The homogeneous orientation in the plane ^{of the} glass plate (i

1) ^{is obtained} by oblique evaporation ^of

SiO at

600-incidence angle [24] ^{on both the} plane surfaces of the glass plate. This latter technique gives ^an uniform director alignment ^{on the whole} plate ^{with a} high anchoring energy. In order to check the director alignment ^at the surfaces of the glass plate

we make

thin nematic layer by sandwiching ^{the nema-}

tic between two parallel glass plates treated in the

same way. The orientation of the director in this layer

is observed by using ^{a Zeiss} polarizing microscope.

Both the surfaces of

single glass plate ^are analysed

in such

way, After, ^the glass plate ^is suspended ^to

the quartz ^{wire and} dipped in the nematic sample,

and the torque

is measured (Fig. 1).

A great ^care is devoted to reduce spurious magnetic torques ^on the torsion pendulum. ^These spurious

torques ^{can be due to small} residual gradients ^{of the} magnetic field, ^to capillarity effects and to the aniso-

tropic shape ^{of the} glass plates ^{P and M of} figure ^1.

For

diamagnetic plate these effects are expected ^to depend ^{on the} square power of the magnetic ^field

The spurious ^{effects are evaluated} by measuring ^the

residual torque ^{exerted on the} glass plate ^{when the}

nematic LC is in its isotropic phase. ^{In our} experiment

the spurious ^torque depends ^{on H in}

^{much more} complex way than ^a simple quadratic form. This suggests ^{the presence of a small amount} of ferroma-

gnetic impurities ^{in the} experimental apparatus. ^This torque depends ^{on the} angle between the glass plate ^P

and the magnetic ^{field A} suitable choice of this angle

allows to reduce the spurious ^{torque to less than 5} %

with respect ^{to the} torque measured in the anisotropic phase. Furthermore the spurious torque ^{does not}

change (within ^our experimental accuracy) ^{when the}

temperature ^{is increased} by ^{more than 15 OC above}

the clearing value. Therefore we can correct the torque measured in the anisotropic phase of the nematic

liquid crystal by subtracting ^the spurious background

measured in the isotropic phase. ^The reliability ^{of this}

latter procedure is checked by looking ^{at the} magnetic field-dependence ^{of the} torque ^{in the} anisotropic phase. ^In fact, uncorrected experimental values show

some small (1-2 %) systematic ^{and not monotonic}

deviations from linearity ^which disappear ^{after cor-}

rection for the spurious background Under this con-

ditions,

estimates that the _presence of the spurious

torque gives

^{residual error}

^the experimental

measurements lower than 1-2 %.

However the _presence of these spurious ^effects

forces ^us to perform ^the experiment by orienting ^the glass plate ^{at the} angle

^which minimizes the spurious

torque. ^{As a} consequence of this the dependence ^of

the surface torque ^{on the} angle ^{cannot be} investigated

in detail. This limits greatly ^the accuracy of the measu- rement of the splay ^{elastic constant as we have} already

shown (Sect. 2).

Accurate and reproducible measurements of elastic constants need an uniform director distortion occurr-

ing ⁱⁿ the nematic sample. This condition can be

accomplished by cooling ^{the LC} starting ^{from the} isotropic phase ^{in the} presence of

high magnetic

field (8.8 kG) ^{which makes an} angle fi :0 ^{90 with}

the easy axis of the director. Figure ^{4 shows the sur-}

face torque ^{measured at the} temperature ^T

^{30 °C}

versus

the p-angle ^{in the} geometry ⁱ

^{1. Each} expe- rimental value of r in figure ⁴ has been obtained after having ^{cooled the} sample ^{from the} isotropic phase ^{in the} presence of the 8.8 kG magnetic ^field According ^{to the} theory (Eq. (11)) the surface torque vanishes when 03B2 ~ ^{900. An} analogous dependence ^of

ï

on the fi-angle is found for the geometry i

^{3 of} figure ^2.

A different behaviour occurs if the magnetic ^{field is}

swiched on when the nematic sample is in the aniso-

tropic phase. ^{In this case} the measured value of the torque ^{is much} smaller than the corresponding ^one

obtained by ^the previous procedure. This behaviour indicates that the director distortion close to the glass plate ^{is not} uniform and domains with opposite ^dis-

tortions of the director-field occur [22]. If fi 0 ^900,

the two states characterized by opposite distortions of the director-field have different free energies. ^There-

fore the distortion which corresponds ^to

higher ^value

of the free _energy is unstable and tends to relax with

time [22]. ^{In order to measure the} relaxation time

from the unstable state to the stable one

have

Fig. ^4.

Dependence of the elastic surface torque

^the angle 03B2 ^between the easy axis and the magnetic ^{field in the} geometry ⁱ

¹ (homogeneous ^director orientation). ^The torque ^is expressed ⁱⁿ arbitrary units. Each experimental point has been obtained after having cooled the nematic

sample ^{from the} isotropic phase in the presence of a 8.8 kG

magnetic ^{field The} temperature ^{of the} sample ^{is T}

^{30 °C.}

prepared ^an uniformly ^distorted sample by cooling

the LC from the isotropic phase ^{in the presence of the}

8.8 kG magnetic ^field for fi

70°. Once the sample

had reached the temperature ^T

30 °C,

^rotated the torsion pendulum ^{to obtain}

p-angle greater than 90°. In this condition the original distortion is not stable and must relax to the stable one. The relaxation of the system ^{can be detected} by measuring ^{the time-} dependence of the surface torque. Figure ^{5 shows}

some relaxation curves obtained by ^this procedure ^for

different values of the p-angle. ^The geometry ^{of the} glass plate corresponds ^{to the i}

^{1 case of} figure ² (homogeneous alignment). ^As expected ^{the free} energy difference between the two director distortions is

increasing function of 03B2-90°, ^{and thus,} the relaxa- tion time TR (see Fig. 5) becomes slower and slower

as fi approaches ^{90°. In} particular ^{we find that} TR diverges (slowing down) ^{when a critical} angle Pc ’" ^93°

is reached For 03B2 Pc the relaxation rate À,R

¹ TR

shows

critical behaviour [4 ^{oc (fi -} 03B2c)]. ^For

90° fi Pc

relaxation is observed.

In the other experimental geometries (cases ^{i = 3}

and i

2), the relaxation from the unstable state does not occur in the whole range of allowable p-angles (70° 03B2 110°). ^In particular ^no change ^{of the}

surface torque ^was observed ^{after one} _day. ^This

suggests ^{that the critical} angle Pc, ^{in these two latter}

cases, is greater ^{than 110°.}

3.2 ELASTIC CONSTANT MEASUREMENTS. 2013 As shown in section 2, ^the torque measurements fumish the

product K33 xa. Therefore the effective _accuracy

Fig. ^5.

Time relaxation from

unstable state to

stable

for différent values of the fl-angle ^{between the}

easy axis and the magnetic ^{field :} (0) fi

^98.250, (*) 03B2 = 96.75°, (*) P

95.25°, (a) fi

93.25°. The measurements

performed ^{in the} geometry ^{i = 1 of} figure ^{2. On the ordinate}

scale is reported the measured value of the torque expressed

in arbitrary units. For convenience the experimental torques corresponding ^to different values of the p-angle ^{have been}

rescaled in such

way

^to coincide at the time t

0.

TR represents the relaxation time. The magnetic ^{field value}

is 8.8 kG and the temperature of the nematic sample ^is

T

30°C.

of the elastic constants depends ^{also on the} accuracy of the diamagnetic anisotropy xa. Analogously ^{in the}

case

of the Freedericksz transition the measurements fumish the ratio K33/~03B1. Measurements of the diama-

gnetic anisotropy ^are often affected by large experi-

mental inaccuracies. Therefore

comparison ^between

the experimental ^{results obtained} by Freedericksz transition experiments ^and by torque measurements can fumish some indications on the reliability ^{of the}

allowable values of _{la. In} the case of 5CB two different measurements of _xa have been reported [25, 26]. ^The

values of X,,, given in reference [25] ^{are - 8} % higher

than those of reference [26]. ^{The elastic constants of}

5CB have been measured by Madhusudana and Pratibha [1] ^and by Bunning et ^al. [27] ^{who used the}

Freedericksz transition technique. ^{The threshold fields}

measured in these papers agree between them within

a

few _per cent. Therefore, ^{in the} following ^{we will refer} only ^to the results of reference [27]. The authors of reference [27] obtained the elastic constants by using

the xa-values given in reference [26]. ^{Different values}

of the elastic constants can be obtained by using ^the

same results of reference [27] ^{but the} magnetic ^aniso- tropy given in reference [25]. ^{In the} following ^{we will}

denote these two sets of elastic constants and ^corres-

ponding diamagnetic anisotropies

^{DATA 1 »} [26, 27] and « DATA II » [25, 27], respectively, ^These

parameters ^are reported ^{in table I.}

The elastic torque per unit surface versus the _magne- tic field in the geometry ⁱ

¹ is shown in figure ^6.

Different symbols correspond ^{to different} tempera-

tures. The linear dependence ^{of r on the} magnetic

field ensures that the anchoring energy coefficient is

Table I.

Splay and bend elastic constants and diamagnetic anisotropies of ^{5CB. Data I and Data II} correspond

to the elastic constants calculated by using ^the experimental Freedericksz thresholdfields given ⁱⁿ reference [27]

and the diamagnetic anisotropies given ⁱⁿ references [26] ^and [25]. ^Our results indicated with and without the _apos-

trophe ^{have been obtained} by using ^the torques ^{measured in the} present experiment ^{and the} diamagnetic anisotropies given ⁱⁿ references [25] ^and [26].

strong enough. Since the relative accuracy ^on the measurement of

is

2 %,

estimate from _equa- tions (12) ^and (13) ^an anchoring energy coefficient greater ^{than 2}

10- 3 erg/cm2 in the whole _range of

nematicity of 5CB. This result is consistent with the value Wo - ²

^10-’ erg/cm2 reported ^{in refe-}

rence [28]. ^The parameter A(l) = 03C4(i = 1) SH ^{can be}

obtained by the best linear fit of the experimental

results of figure ^{6. The} temperature dependence ^of A(1) is shown in figure ^{7. The full and broken curves}

in figure ⁷ correspond ^to the theoretical torques calculated by substituting ⁱⁿ equation (13) ^{the sets}

of data I and II, respectively.

The elastic torque per unit surface versus the

magnetic ^{field H in the case i}

3 is shown in figure ^8.

In this case, too, the linear dependence ^{of 03C4 on the} magnetic ^field indicates the anchoring of the director at the surface is strong enough. ^In particular ^{we can}

estimate that the anchoring ^energy coefficient for the

homeotropic alignment ^{due to RBS is} greater ^than

5

10- 3 erg/cm. The best linear fit of the experimental

results of figure ^{8 allows us to} obtain the parameter A 3 = 03C4(i = 3) SH. The température dependence of A(3)

is shown in figure 9. The full and broken lines cor-

respond ^{to the} theoretical values obtained by ^substi- tuting ⁱⁿ equation (13) ^{the two sets of data I and II,} respectively. Notice that in both the geometries ^{i = 1}

and i

3 the measured torques lie between the full and broken curves. Therefore our results _agree with those of reference [27] ^{if one accounts for the uncer-} tainty ^{on the} diamagnetic anisotropy. Furthermore

our

results suggest ^{that the correct value of the}

diamagnetic anisotropy is intermediate between those

given in references [25] ^and [26].

The experimental ^values of il ^and K33 ^{can be}

obtained from A(1) ^and A(3) by using ^the procedure already discussed in section 2. The values of K11

and K33 obtained in this _way ^are reported in table I.

The elastic constants with and without the apostrophe

have been obtained by using ^the diamagnetic ^aniso- tropies given in references [25] ^and [26], respectively.

As shown in section 2, ^the accuracy of K11 ^{is poor} (~ 50 %), whilst the _accuracy of K33 is estimated of the order of 20 %.

More accurate values of K33 (~ ¹⁰ %) ^{are obtained} by using equation (13) together with the values of 1

measured by other methods [27]. The bend elastic constant K33 ^{can be obtained} by substituting ⁱⁿ equation (13) ^the experimental values of either A(1)

or A(3). Comparison between the values of K33

obtained from these ^two independent measurements

gives ^a direct check of the _accuracy of our experimental

results. Figure ^{l0a shows our} experimental ^{values of} K33

^the temperature ^as deduced from A(l).

Full circles and _open circles correspond ^{to the values}

of K33 ^obtained by substituting ⁱⁿ equation (13)

Fig. ^6.

Torque for unit surface

in the

i

1

(ç

90°)

^the magnetic field Different symbols correspond ^to different values of the temperature : (A) Tc - ^T

13.4°C, (*) Tc - ^T

10.4°C, (0) TT. - ^T

8.4°C, (*) Tc - ^T

⁶ oC, Tr - ^T

^3.8 oC, (9) Tc - T = 2.2 °C, (D) T, - T = 1.2°C,(~)Tc - T ^=0.28°C.

The full lines represent the best linear fit of the experimental

results. The value of the a-angle ^{between the} y-axis ^{and the}

axis orthogonal ^{to the} glass plate ^{is 87.3°} (see Fig. 3).

Fig. ^7.

Temperature dependence ^of A(1)

03C4(i = 1) SH.

The full and broken

correspond ^{to the} expected

values of A(1)

^obtained by substituting ⁱⁿ equation (11)

the two sets of experimental data denoted

I and II,

pectively (see discussion in the text). The value of the angle

is

87.3°.

Fig. ^8.

^Elastic torque per unit surface

the

magnetic ^{field in the case i = 3} (cp = 0°) for different values of the temperature : (Ã) Tr - ^T

13 °C, (0) Tr - ^T

9.9 oC, (A) Tr - ^T

8 °C, (0) Tr - ^{T = 5.5} °C, (0) Tc - ^T

3.4°C, (*) Tc - ^T

1.8°C, (*) Tr - ^T

0.8 °C. The full lines correspond ^to the best linear fit of the

experimental results. The value of the a-angle ^{is 10.6°.}

Fig. 9. - Température dependence ^of A(3) = 03C4(i = 3) SH.

The full and broken

correspond ^{to the} theoretical torques calculated by using ^the

procedure ^described

in figure 7. The value of the angle

^{is 10.b°.}

the ~03B1 ^values given in references [26] ^and [25], respec-

tively. The full and broken curves correspond ^{to the}

elastic constants K33 ^{denoted as data 1 and II,} respec-

tively. Figure lOb shows the values of K33 ^obtained

by using ^the experimental ^{values of} A(3).

Fig. ^10.

a) Experimental ^values of K33

^{obtained from}

surface torque measurements in the

of the geometry

i

1 of figure ^2. b) Experimental ^{values of} K33 ^obtained

from torque measurements in the geometry ⁱ

^{3. Full} circles and open circles represent

experimental ^values

of K33 ^obtained by using the values of ~03B1 given ^{in refe-}

rences

[26] ^and [25]. The full and broken lines correspond ^to

the experimental ^{values of} K33 ^denoted

data 1 and data II in table I, respectively.

4. Conclusions.

Experimental measurements of the torque ^exerted by

nematic LC on an interface are shown to furnish a new method to measure the elastic constants of nematic LC. In the case of weak anchoring ^{of the}

director at the interfaces, ^the anchoring energy ^too

can be obtained in an unambiguous way from surface torque measurements. Note that the torque ^method

can also be used to measure the anchoring energy when the director alignment ^at the interface is tilted A direct measurement of the anchoring energy coefficient at a SiO-nematic interface obtained by using the elastic torque method has been given by

us in reference [23]. ^The torque measurements

practically unaffected from the typical error sources

affecting ^the

standard » Freedericksz transition

thod (director misalignment ^{at the} surfaces, ^weak anchoring energy). ^{However, the} torque measurements are more complex ^{and the} splay ^{elastic constant}

cannot be actually obtained with a satisfactory

accuracy. A great improvement ^{of the} torque ^method could be obtained if the spurious angle-dependent background ^was suitably ^{reduced. In this case both}

the bend and splay ^{elastic constants} could be obtained with

better accuracy from measurements of the

angular dependence of the elastic torque ⁱⁿ

single experimental geometry (i ^{= 1}

ⁱ

^{3 in} Fig. 2).

In our opinion ^the torque technique ^{finds its best} application ^{in the} measurement of the twist elastic constant K22 of nematic LC. In fact the optical

measurements of K22 from the Freedericksz threshold

are particularly ^difficult and affected by large experi-

mental errors [1]. ^{On the} contrary ^the measurements of the twist elastic constant by the torsion pendulum

are

the simplest ^{ones since} they ^concem

single experimental geometry (i

^{2 in} Fig. 2) [17].

An _open question ^{concerns the role} played by ^the

surface-like elastic constant introduced by Nehring

and Saupe [20]. ^The satisfactory agreement ^between

our

éxperimental results and the theoretical predic-

tions obtained by assuming K13

⁰ (Figs. ^{7 and} 9)

seem to indicate that Kt3 ^{does not} play ^an important

role when the geometries ^of figure ^{2 are} investigated

However a definitive _response to this question ^cannot

be reached before

full theory of the surface elasticity

is developed [21].

In conclusion, although ^the present technique

cannot replace the Freedericksz transition method,

it

constitute

useful alternative _way ^to measure

the twist and bend elastic constants and the anchoring

energy of liquid crystals. Furthermore,

great ^increase of the experimental accuracy of this method ^can be

expected if some technical difficulties will be overcom-

ed in the future work.

References

[1] MADHUSUDANA, ^{N. V. and} PRATIBHA, R., ^Mol. Cryst.

Liq. Cryst. ⁸⁹ (1982) ^249.

[2] ^SCHAD, Hp. ^and OSMAN, ^M. A., J. Chem. Phys. ⁷⁵ (1981) 880.

[3] ^{DE JEU, W. H., CLAASSEN, W. A. P. and SPRUIJT,}

A. M. J., ^Mol. Cryst. Liq. Cryst. ³⁷ (1976) ^277.

[4] HALLER, I., ^{J. Chem.} Phys. ⁵⁷ (1972) ^1400.

[5] LEENHOUTS, F., ^VON

WOUNDE, ^{F. and} DEKKER, A. J., Phys. ^{Lett. 58A} (1976) ^242.

[6] OLDANO, C., MIRALDI, E., ^TAVERNA VALABREGA, P.,

J. Physique ⁴⁵ (1984) ^{755 and}

OLDANO, C., MIRALDI, E., STRIGAZZI, A., ^TAVERNA-